Theorem equsexd 1657

Description: Deduction form of equsex 1656. (Contributed by Jim Kingdon, 29-Dec-2017.)

Hypotheses

Ref	Expression
equsexd.1	|- ( ph -> A. x ph )
equsexd.2	|- ( ph -> ( ch -> A. x ch ) )
equsexd.3	|- ( ph -> ( x = y -> ( ps <-> ch ) ) )

Assertion

Ref	Expression
equsexd	|- ( ph -> ( E. x ( x = y /\ ps ) <-> ch ) )

Proof of Theorem equsexd

Step	Hyp	Ref	Expression
1		equsexd.1	. . 3 |- ( ph -> A. x ph )
2		equsexd.2	. . 3 |- ( ph -> ( ch -> A. x ch ) )
3		equsexd.3	. . . 4 |- ( ph -> ( x = y -> ( ps <-> ch ) ) )
4		bi1 116	. . . . 5 |- ( ( ps <-> ch ) -> ( ps -> ch ) )
5	4	imim2i 12	. . . 4 |- ( ( x = y -> ( ps <-> ch ) ) -> ( x = y -> ( ps -> ch ) ) )
6		pm3.31 258	. . . 4 |- ( ( x = y -> ( ps -> ch ) ) -> ( ( x = y /\ ps ) -> ch ) )
7	3, 5, 6	3syl 17	. . 3 |- ( ph -> ( ( x = y /\ ps ) -> ch ) )
8	1, 2, 7	exlimd2 1526	. 2 |- ( ph -> ( E. x ( x = y /\ ps ) -> ch ) )
9		a9e 1626	. . . 4 |- E. x x = y
10	1	a1i 9	. . . . . . . . 9 |- ( ph -> ( ph -> A. x ph ) )
11	10, 2	jca 300	. . . . . . . 8 |- ( ph -> ( ( ph -> A. x ph ) /\ ( ch -> A. x ch ) ) )
12		prth 336	. . . . . . . 8 |- ( ( ( ph -> A. x ph ) /\ ( ch -> A. x ch ) ) -> ( ( ph /\ ch ) -> ( A. x ph /\ A. x ch ) ) )
13	11, 12	syl 14	. . . . . . 7 |- ( ph -> ( ( ph /\ ch ) -> ( A. x ph /\ A. x ch ) ) )
14		19.26 1410	. . . . . . 7 |- ( A. x ( ph /\ ch ) <-> ( A. x ph /\ A. x ch ) )
15	13, 14	syl6ibr 160	. . . . . 6 |- ( ph -> ( ( ph /\ ch ) -> A. x ( ph /\ ch ) ) )
16	15	anabsi5 543	. . . . 5 |- ( ( ph /\ ch ) -> A. x ( ph /\ ch ) )
17		idd 21	. . . . . . . 8 |- ( ch -> ( x = y -> x = y ) )
18	17	a1i 9	. . . . . . 7 |- ( ph -> ( ch -> ( x = y -> x = y ) ) )
19	18	imp 122	. . . . . 6 |- ( ( ph /\ ch ) -> ( x = y -> x = y ) )
20		bi2 128	. . . . . . . . 9 |- ( ( ps <-> ch ) -> ( ch -> ps ) )
21	20	imim2i 12	. . . . . . . 8 |- ( ( x = y -> ( ps <-> ch ) ) -> ( x = y -> ( ch -> ps ) ) )
22		pm2.04 81	. . . . . . . 8 |- ( ( x = y -> ( ch -> ps ) ) -> ( ch -> ( x = y -> ps ) ) )
23	3, 21, 22	3syl 17	. . . . . . 7 |- ( ph -> ( ch -> ( x = y -> ps ) ) )
24	23	imp 122	. . . . . 6 |- ( ( ph /\ ch ) -> ( x = y -> ps ) )
25	19, 24	jcad 301	. . . . 5 |- ( ( ph /\ ch ) -> ( x = y -> ( x = y /\ ps ) ) )
26	16, 25	eximdh 1542	. . . 4 |- ( ( ph /\ ch ) -> ( E. x x = y -> E. x ( x = y /\ ps ) ) )
27	9, 26	mpi 15	. . 3 |- ( ( ph /\ ch ) -> E. x ( x = y /\ ps ) )
28	27	ex 113	. 2 |- ( ph -> ( ch -> E. x ( x = y /\ ps ) ) )
29	8, 28	impbid 127	1 |- ( ph -> ( E. x ( x = y /\ ps ) <-> ch ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   A.wal 1282   = wceq 1284   E.wex 1421

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-i9 1463  ax-ial 1467

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  cbvexdh  1842